Are "not uncommon" and similar phrases double negatives? Should their use be avoided?

To answer your first explicit question, I would say they are double negatives:

A double negative occurs when two forms of negation are used in the same clause.

To answer the second question, I would say the use of litotes is perfectly acceptable.

Litotes is a form of understatement, always deliberate and with the intention of emphasis. However, the interpretation of negation may depend on context, including cultural context. In speech, it may also depend on intonation and emphasis; for example, the phrase "not bad" can be said in such a way as to mean anything from "mediocre" to "excellent."

The respective Wikipedia articles (linked to and excerpted above) give a lot of good information. I would like to emphasize the potential ambiguity in litotes, in that the intensity of the double-negative-as-positive ranges from "mildly positive" to "resoundingly positive".

Finally, see this other EL&U question covering the specific example of not uncommon.

It's a rhetorical technique called litotes, and it goes back at least as far as Homer:

οὔτε γάρ ἔστ᾽ ἄφρων οὔτ᾽ ἄσκοπος... ("he is neither unthinking, nor unseeing")

If it's good enough for St. Paul ("I am a Jew, from Tarsus in Cilicia, a citizen of no ordinary city" -- Acts 21:39) it's good enough for, well, you.

EDIT I forgot my favorite example, Harold and Kumar Go To White Castle, which contains several exchanges like

Kumar: You're worthless

Harold: (dejected) I'm not worthwhile

John Y. points out that not all litotes fit the OP's model of "not un-". In fact, the best (St. Paul, John Cho, Sade) don't. I think it's, ahem, not unreasonable to say that the "not un-" style can sound plodding and even pompous.

(While I'm wondering off topic: none of this should be interpreted as defense, even by way of faint condemnation, of the real double-negation of the "we don't need no education" variety, which is Just Plain Wrong.)

IMHO, no, it is not a double-negative. At least not the normal sense of the word. A double-negative refers to using two negatives, as in not and no, when trying to convey one meaning where you actually end up saying the opposite. "It is not uncommon" is a negation of a word with a negative meaning.

Think of it this way. Can you say "I'm not going to do no homework" in another way that makes your meaning clearer? Obviously you can. "I'm not going to do any homework." Can you say "It is not uncommon" in a way that makes your meaning any clearer? Nothing comes to mind for me. You could say, "It is ordinary", but that has a slightly different meaning to me.

Having two negation terms together intended as negation is common in colloquial and non-standard dialects of English, but it is prescribed against in Standard English. For example:

There ain't no chitlins left in the skillet.

which translated to Standard English is

There are no fried pig intestines remaining in the frying pan.

or

There aren't any fried pig intestines remaining in the frying pan.

(ignore for the moment that "ain't" is also prescribed against in SAmE)

This is primarily what one (your school teacher) referred to when they said "Double negation is forbidden".

But using two negatives in order to create a logical positive, (or weak non-negative), while cognitively laborious, is perfectly allowable. The phrase

not uncommon

...is 'legal' English and is intended to mean common or rather more accurately "somewhat more frequent than rare, i.e. not rare, but I don't want to go so far as to say frequent".

So one can use two negatives when the intention is to mean a positive.

So for your examples, the first one is OK in some varieties of non-standard English (but of course grates, as you said, in Standard English). Your second example is -not wrong- just slow to make sense of and most likely was intended to mean "I will go to Greenland".